Introduction to Concept Space

science, philosophy

Maps and Metaphors

I’m currently reading a book called Metaphors we Live By by Lakoff and Johnson, and it’s probably one of my favorite things I’ve read all year. I’ve been working through it for a few weeks now, and while it’s not a hard book, I find myself needing to go really slowly, namely because it has a super-high insight-density and I need to keep stopping and actually thinking about the things it says. Seriously, this book has literally changed the (figurative) way I see the world, and if you’re looking for such a book, I would highly recommend it.

While the authors don’t necessarily come out and say it in so many words, one of the book’s concepts is an axiomatic yet reducible set of theorems as the basis of human understanding. The theorems are axiomatic in the sense that they can be used to derive all other concepts – that every possible idea we can think of is (arguably) made up of some combination of these – but reducible in the sense that we can perform introspection on them and break them down.

“Well,” you might be starting to ask, “aren’t the things they decompose to then the axioms, not this set?” And indeed, you would be correct on that front in most cases, except here, because the ideas they reduce to are other ideas in the set of axioms.

If this seems counterintuitive to you, think about it like this: each of the axioms is defined in terms of the other axioms. The notion will probably make more sense if we stop thinking of the axioms as theorem-components and instead as the basis of a vector space. If you’ve forgotten, a vector space (very loosely) generalizes coordinates.

As a quick example, imagine a map of the world. It’s obvious that we have only two dimensions on this map – the X and the Y coordinates. Furthermore, we also have the notion of distance between any two places on the map. If you’re not a mathematician, this is (hopefully) a good enough approximation of a vector space to follow along with the remainder of my argument. That being said, remember that vector spaces can have as many dimensions as they want. Our map has 2, but we will shortly see an example of a 7-dimensional space, and we could have infinite dimensions if we really wanted.

Keeping our map example in mind, a basis is any convention for measuring locations in the space. By historical convention, we generally use a basis of [latitude, longitude] for expressing locations on the planet, which roughly corresponds to [North, West]. We use North and West (or South and/or East), because they are vectors (mathematical objects with the notion of a direction) which happen to be orthogonal – which is to say, point in directions that are 90 degrees away from one another. It is completely impossible to point North if you are only able to point East or West.

However, [N, W] is not the only basis we can choose for our map. You can come up with some trivial other bases to choose from, but one you almost certainly didn’t think of is the basis [NE, NEN] – northeast and northeast-north. It’s not as easy as using [N, W], since moving in the NE direction also moves you in the NEN direction, but if you try, you actually can describe any location with a unique set of coordinates. On a map, you can pick any two separate cardinal directions to form a basis, so long as they don’t point in completley opposite directions.

This idea indeed generalizes to vector spaces at large, with a few small changes: in order to describe an N-dimensional vector space, we are going to need a basis with N different vectors, and none of those vectors must be entirely constructable as a combination of any of the others.

The intuition here is that for any vector space, our choice of basis is arbitrary, and serves to describe any location in the entire space. If you get to choose, pick an orthogonal basis (where every vector in the basis is orthogonal to every other vector in the basis), because it makes your life easier, though this isn’t strictly necessary to actually do anything cool.

All of a sudden, the notion that concept-space be made up of reducible axioms need not seem so strange anymore. If we look at concept-space as a vector space, all we are expressing is that it have a basis which is non-unique.

Metric-Space is a Metric Space

To better hammer-in what concept-space might look like, let’s take a simpler toy example to knock out some of the more subtle details.

If you are not American, you will be very familiar with the metric system. If you are American, you should consider switching, because your system is stupid.

At its core, the metric system is composed of seven fundamental units: mass (kg), time (s), distance (m), count (mol), electrical current (A), temperature (K) and luminance (cd). At first glance, this seems incapable of expressing most of the things we want to measure, like speed, volume (the liquid sense), or power.

However, as physicists or otherwise Johnny-on-the-spot readers might see, all physical measurements turn out to be combinations of the seven listed previously. Speed is just \(m\cdot s^{-1}\), volume \(m^3\), and power \(kg\cdot m^2 \cdot s^{-3}\).

It is evident, then, that the seven base metric units form a basis for the metric system. We can describe any physical measurement as a 7-tuple of [kg, s, m, mol, A, K, cd], analogously to how we could describe any location on a map of the world as a 2-tuple of [N, W]. Furthermore, to expand our metaphor, we can look at metric-space itself as a map of the measurements, where each individual unit has a definite location in metric-space, just like every city has a definite location on Earth.

Also it is important to note that choice of basis for the metric system (the seven base units) be just one of an infinite number of possible bases. Just like we could describe the map’s coordinates with the [NE, NEN] basis, we can construct a basis for the metric system by using seven derived units of measurement – so long as no two are the same and none can be made up by the other six. It is left as an exercise to the reader to find a basis for the metric system that satisfies these constraints.

Word-Space: Not Just a Space of Words

Despite being a software engineer by trade, I’m always amazed by the incredible things people seem to be able to coerce computers to do. There’s a program called word2vec that has recently blown my mind more than most.

word2vec does this: given a large-enough corpus of text, builds a vector space1 that encodes the relationships between words. From the manual:

It was recently shown that the word vectors capture many linguistic regularities, for example vector operations vector(‘Paris’) - vector(‘France’) + vector(‘Italy’) results in a vector that is very close to vector(‘Rome’), and vector(‘king’) - vector(‘man’) + vector(‘woman’) is close to vector(‘queen’)

In other words, this software mathemagically can determine that removing the idea of [man*]* (square-bracket notion explained here) from [king] leaves some sort of abstract notion of [royalty], such that we can add it to [woman] and get [queen].

Also in the manual but not mentioned in the above excerpt is that taking the vector closest to [river] + [China] results in [Yangtze].

Computers did that, with absolutely no philosophical “understanding” of the text being analyzed. Frickin’ computers! Please take a second to update on this information and move your priors a little closer towards [there is nothing magical about human intelligence]. This is kind of the exact reason why I’m in this field.

An important point here is that – as far as I can tell – the underlying vectors here (which is to say, the actual coordinates in word-space) have no real meaning. However, once you slap some semantic labels on the vectors you care about, the semantic relationships we seem to care about are preserved through manipulation of the vectors. The coordinates are nothing but the mathematical artifacts responsible for making this work “behind the scenes”. However, I will leave the (inevitable) philosophical debate about the “meaning of these coordinates” to the philosophers. To me, the coordinates are just linear combinations of eigenvectors. But I digress.

A subtle point is that when discussing the [Yangtze], we talked about “the closest vector”. It is a fundamental property of vector spaces (and more generally, metric spaces (not to be confused with metric-space: the vector space of the metric system)) that, just like cities on a map, vectors with similar coordinates are close to one another in space. The reason I bring this up is to assuage your fears that perhaps it isn’t meaningful to discuss the concept of “distance” when talking about highly abstract ideas. Good news: if you can prove you’re talking about a vector space, it is!

In practice, what this means is that the ideas [man], [king], [lion] and even [Budweiser] are all spatially-located pretty closely to one another in concept-space, though likely in different dimensions. Just like how the longitude and latitude of any two points on a mountain are likely to be pretty similar, they can have wildly varying altitudes, [man] and [king] are separated only by the notion of [royalty], while [king] and [lion] might be separated only by the concept [feline]. Ideas which are located closely with one another, in aggregate form what is known as a cluster – clusters in concept-space are groups of similar ideas.

The Concept of Concept-Space

Warning: abandon all hope of mathematical rigor, ye who enter here.

While thinking about the word-space as created by word2vec is cool and all, it seems to me like we should be able to do better. Word-space is descriptive – which is to say it is generated BY the ideas we currently have – rather than prescriptive, which would ideally allow us to postulate entirely new concepts by just adding constituent vectors together.

The remainder of this essay should explicitly be classified with an epistemic status of “almost certainly false, but potentially a useful tool for modeling regardless”. Proceed at your own risk.

Definitions

Assume there exists a meaningful concept-space – a vector space of every thought possibly thought. This implies every thought has some definite coordinate representation in concept-space, even if we’re unsure about what the coordinates actually mean (as in the case of word-space). For simplicity, we shall call points in concept-space thoughts. Furthermore, we assume it be meaningful to combine thoughts with one another (that a combination of two thoughts results in another point in concept-space).

With the assumption that concept-space have finite degree (though the argument doesn’t depend on this assumption, it makes our terminology relatively-consistent with existing literature), let any basis of concept-space be called a prior, in adherence with the Bayesians.

Let’s say that at any given moment, you are able to recall some set of ideas you’re previously thought. For the most part, arguably this set of ideas uniquely defines your personality. This encompasses the topics you like talking about, the explicit affordances you have to external stimulus, and the thought patterns your mind takes when tackling new problems. For lack of a better term, I will call this your thought \(bucket\).

By an argument of diagonalization, you should be able to combine two of the thoughts in your \(bucket\) to create a new thought, one which is not currently in your \(bucket\) (though this argument doesn’t hold if one of the thoughts be the additive identity). This process allows us to come up with thoughts previously un-thought, arguably the processes of creativity and problem-solving. Being able to generate new ideas does not necessarily imply that you can generate good ideas, however.

An Insight into Insights

Almost by definition, then, the majority of the thoughts in your \(bucket\) will cluster with one another, since they have been composed of one another. We can thus define an insight (in the usual sense of the word) as a thought with a large distance between itself and the weighted center of the \(bucket\) cluster. Suddenly, our model gains predictive power, in that one insight begets another: adding this new insight to any other thought in the \(bucket\) will likely lead to another insight. Indeed, it is generally considered the case that insights lead to more insights[citation needed].

Exploration of Concept-Space

Let us call this act of trying to find specific thoughts the exploration of concept-space. Since our model describes exploration as the linear combination (and insertion henceforth) of thoughts from (to) the \(bucket\), this process is necessarily serial. Let us then define the difficulty of an arbitrary thought \(T\) to be the fewest number of combinations of thoughts from the \(bucket\) necessary to compose thought \(T\). (Mathematically speaking, this difficulty is defined as the minimal sum of the norms of the coefficients of \(T\) when expressed in the basis defined by \(bucket\).)

Clearly our choice of prior for concept-space will determine the difficulty of any given thought. If looking for a particular thought, the best case is that you already know it; the second best case is if it’s close to something you already know; the third best case being if you have lots of thoughts completely unrelated to one another, allowing you to quickly explore areas of concept-space you’ve never been to before.

Intuitively, the fact that difficulty is a function of prior describes why some people are better at certain types of mental tasks than others, and why brainstorming works better in a group (the group can combine their individual \(bucket\)s, and thus search arbitrary sections of concept-space more efficiently).

The Effects of Priors on Exploration

The takeaway here is that not all priors (read: ways of thinking) are created equally. Different priors are more effective for exploring different regions of concept-space, analogously to how some vehicles are better suited to getting to some places.

Assuming complete indifference of concept-space, the best (difficulty-minimizing) prior is orthogonal: each thought vector points in a completely different direction than every other thought (recall that North and East and Up are all orthogonal on a map). If you want to be able to get to anywhere in concept-space least-difficultly, it is best that your prior express no bias whatsoever. An indifference prior manages an average-level difficultly for all thoughts at the expense of a complete inability to find any thought efficiently.

However, just like priors aren’t created equally, neither are thoughts. Some thoughts (eg. [evolution by natural selection]) are simply more useful than others. An indifference prior is, by nature, unable to express such preferences. Clearly we can do better than an indifference prior.

The ideal prior for exploring useful regions of concept-space we shall call the eigenprior, and while it is outside the scope of this paper to find such a prior, the general strategy is principle component analysis (for an awesome example, this article shows how to apply the procedure for facial recognition – pictures included!).

It is an open question whether or not the eigenprior is effectively installable onto human-mindware (let alone if this question is even meaningful).

Conclusion

I’ve got a lot more to write on the topic of concept-space, but this post is already heavily tangential and long-winded, so I’ll save it for another time.

In short, however, the key takeaways from this post I plan to extend on are:

  • concept-space is a vector space, and as such, all of linear algebra applies in its analysis.
  • it is meaningful to discuss the distance between thoughts.
  • any arbitrary thought has an associated difficulty of being found, defined by characteristics of the mind-doing-the-exploration.
  • as such, some means of thinking about the world are simply more useful than others, and it is thus worthwhile to spend some time trying to optimize this process.

In addition, if we’re going to continue thinking about this, it’s going to need a snazzy name. I kind of like “Concept Theory” for its high-degree of general abstract nonsense, but I’m certain that the internet can do better.

Prima facie, this seems like a highly-worthwhile field of investigation – I’d love to hear all of your thoughts.


  1. Well, not really, but for our purposes let’s just pretend.